a tour of pointer analysis - korea...

A Tour of

Pointer AnalysisPointer Analysis

Ondřej Lhoták

Pointer analysis

A Tour of Pointer Analysis| Ondrej Lhotak| University of Waterloo

[PASTE 2001]

What does pointer analysis do?

For each pointer (reference) in the program, what

memory locations (objects) does it point to?

p


q

Why pointer analysis?

a = 1

b = 2

c = a + b 3



a = 1

b = 2

*x = 4

c = a + b ?



a = 1

b = 2

*x = 4

c = a + b ?


If x == &a, then c = 6.

If x == &b, then c = 5.

If x != &a && x != &b, then c = 3.


a = 1

b = 2

foo()

c = a + b

void foo() { ... }



a = 1

b = 2

x.foo()

c = a + b

class Y extends X { class Z extends X {

class X {

void foo() { ... }

}

?

?

?


class Y extends X {

void foo() { ... }

}

class Z extends X {

void foo() { ... }

}

?

Applications of pointer analysis

• Call graph construction

• Dependence analysis and optimization

• Cast check elimination

• Side effect analysis

• Escape analysis• Escape analysis

• Slicing

• Parallelization

• ...


Pointer analysis as an abstraction



p


q




p


q

α( ) =




p


q

α( ) =α( p ) = p


p

q

p

q

Concrete program execution Abstract analysis


α( ) =

q

α( p ) =

q

p

Precision of points-to sets

{L1} ⊂ {L1, L2} ⊂ {L1, L2, L3} ⊂ {L1, L2, L3, L4}

←more precise less precise →

uncomputableunsound conservative


actual

behaviour

on all

executions

actual

behaviour

on some

executions

possible analysis results

Precision vs. efficiency

Pre

cise

CFA trend


← Efficient

Pre

cise→

Theory


Pre

cise


← Efficient

Pre

cise→


Pre

cise


← Efficient

Pre

cise→

Implementation tuning

Precision for a specific application

{L1, L2} ⊂ {L1, L2, L3, L4}

This points-to set is more precise because it is smaller.

But suppose a particular application only cares whether

L1 is in the set. Then for that application, both sets are


L1 is in the set. Then for that application, both sets are

equally precise.

Thus, precision/efficiency tradeoff must consider the

application.

Design decisions for precision/efficiency

•The abstraction (affects precision and efficiency):–Type filtering

–Field sensitivity

–Directionality

–Call graph construction

–Context sensitivity

–Flow sensitivity–Flow sensitivity

•Algorithm and implementation (affects efficiency)–Propagation algorithm

–Set implementation


An example abstraction and analysis

•The abstraction:–Type filtering


–Directionality



–Flow sensitivity

First example:–without type filtering

–field-sensitive

–subset-based

–ahead-of-time call graph

–context-insensitive

–flow-insensitive–Flow sensitivity


–flow-insensitive

Pointer Assignment Graph abstraction

L1

Abstract object node:


Java:

L1: x = new Object()

C:

L1: x = malloc(42)

Represents some set of run-time objects (targets of pointers).

e.g. all objects allocated at a given allocation site

e.g. all objects of a given dynamic type


&a

Address-of abstract object node:


Represents some set of run-time objects (targets of pointers).

e.g. the object whose address is &a.

C:

x = &a


p

Pointer variable node:

&aL1pt(p) = {L1, &a}

Points-to set


Represents some pointer-typed variable(s).

e.g. all instances of the local variable p in method m.


p.f

Pointer dereference node:

*p


Java:

y = p.f

C:

y = *p

Represents a dereference of some pointer

(where the pointer is a pointer variable node).


L1.f

Heap pointer node:

*L1

&aL2L2 L3pt(L1.f) = {L2, L3}


Represents a pointer stored in some object in the heap.

State space (the analysis result)

L1.f

ppt( ) = { , , }L1 L2 &q

pt( ) = { , , }L1 L2 &q


pt : (Var ∪ (Obj × Field)) →℘(Obj)

pt : (Var × Obj) ∪ (Obj × Field × Obj)

where Obj = Alloc ∪ AddrOf

State space (the analysis result)

L1.f

ppt( ) = { , , }L1 L2 &q

pt( ) = { , , }L1 L2 &q


allocation L1: x = new Object()

assignment x = y

Pointer Assignment Graph edges

L1 x

y x

store y.f = x

load x = y.f


y.f x

y.fx

Examplestatic void foo() {L1: p = new O();

q = p;L2: r = new O();

p.f = r;t = bar( q );

}

static O bar( O s ) { return s.f;


return s.f;}

Example

L2

p

p.f

L1static void foo() {L1: p = new O();

q = p;L2: r = new O();

p.f = r;t = bar( q );

}


q

r

s.f


return s.f;}

s

s.f

t

Generate points-to assignment graph.

Example

L2

p

p.f


q = p;L2: r = new O();

p.f = r;t = bar( q );

}


L1

q

r

s.f


return s.f;}

s

s.f

t

Propagate points-to sets.

Example

L2

p

p.f


q = p;L2: r = new O();

p.f = r;t = bar( q );

}


L1

qL1

r

s.f


return s.f;}

s

s.f

t


Example

L2

p

p.f


q = p;L2: r = new O();

p.f = r;t = bar( q );

}


L1

qL1

r

s.f


return s.f;}

sL1

s.f

t


Example

L2

p

p.f


q = p;L2: r = new O();

p.f = r;t = bar( q );

}


L1

qL1

rL2

s.f


return s.f;}

sL1

s.f

t


Example

L2

p

p.f


q = p;L2: r = new O();

p.f = r;t = bar( q );

}


L1

qL1

rL2

s.f

L1.f


return s.f;}

sL1

s.f

t

Add load/store edges.

Example

L2

p

p.f


q = p;L2: r = new O();

p.f = r;t = bar( q );

}


L1

qL1

rL2

s.f

L1.fL2


return s.f;}

sL1

s.f

tL2

L2

Re-propagate points-to sets.

Overall algorithmrepeat until no change {propagate abstract objects along edgesfor each load/store, add indirect edges to heap ptr nodes

}

add all allocation nodes to worklistwhile worklist not empty {remove node v1 from worklistfor each edge v1 -> v2, propagate pt(v1) into pt(v2)

if v2 changed, add v2 to worklist

Simple:

Detailed:


if v2 changed, add v2 to worklistfor each load v1.f -> v3 {

for each a in pt(v1) {add edge a.f -> v3 to assignment graphadd node a.f to worklist

}}for each store v3 -> v1.f {

... (as above)}

}

Comparison with 0CFA

Field-sensitive subset-based points-to analysis:

pt : (Var ∪ (Obj × Field)) →℘(Obj)

0CFA:

⇒ Σ = Call × (Var→℘(Lam))



p

L1: new

p

L1: λ


pL1

pL1

Set implementation• hash: Using java.util.HashSet

• array: Sorted array, binary search

• bit vector:

• hybrid:

a b d g

a b c d e f g h i j

1 1 0 1 0 0 1 0 0 0

• hybrid:– array for small sets

– bit vector for large sets

• sparse bit vector:

• binary decision diagram:


0 (ab) 1 1

1 (cd) 0 1

3 (gh) 1 0

Set implementationhash slow large

array slow small

bit vector fast large

hybrid fast small

sparse bit vector fast small

binary decision diagram depends depends


Slow vs. fast: up to 100x difference

Large vs. small: up to 3x difference

Set implementation is very important.

Incremental propagation

pL1 L2 L3 L4

q



• 1st iteration: propagate {L1, L2, L3, L4}

pL1 L2 L3 L4

qL1 L2 L3 L4




• add L5 to pt(p)

pL1 L2 L3 L4 L5

qL1 L2 L3 L4

• add L5 to pt(p)




• add L5 to pt(p)

pL1 L2 L3 L4 L5

qL1 L2 L3 L4 L5

• add L5 to pt(p)

• 2nd iteration: propagate {L1, L2, L3, L4, L5}



pL1 L2 L3 L4

Idea: Split sets into old part and new part.

new

old

qnew

old



• 1st iteration: propagate {L1, L2, L3, L4}]

pL1 L2 L3 L4


new

old

qL1 L2 L3 L4new

old




• flush new to old

p


L1 L2 L3 L4

new

old

q

L1 L2 L3 L4

new

old






pL5


L1 L2 L3 L4

new

old

q

L1 L2 L3 L4

new

old


• add L5 to new part of pt(p)





pL5


L1 L2 L3 L4

new

old

qL5

L1 L2 L3 L4

new

old



• 2nd iteration: propagate {L5}





p


L1 L2 L3 L4 L5

new

old

q

L1 L2 L3 L4 L5

new

old



• 2nd iteration: propagate {L5}






–Directionality







Type filteringA x, z;B y;

A: x = new A();B: y = new B();

y = (B) x;z = y;

x

A

y

B


Inheritance hierarchy:

A

B C

z

Type filtering: none A x, z;B y;


y = (B) x;z = y;

x

A

y

B

A B

A



A

B C

zA B

Type filtering: after analysisA x, z;B y;


y = (B) x;z = y;

x

A

y

B

A B

A



A

B C

zA B

Type filtering: during analysisA x, z;B y;


y = (B) x;z = y;

x

A

y

B

A B

A



A

B C

zA B

Type filtering: effect on heap nodesclass A {Object fA;

}class B {Object fB;

}class C {Object fC;

}A: a = new A();

A.fA B.fA C.fA

A.fB B.fB C.fB

A.fC B.fC C.fC


A: a = new A();B: b = new B();C: c = new C();

Type filtering

• Ignoring types yields many large points-to sets.

• Filtering after propagation is almost as precise as

during propagation.

• Filtering during propagation is both most precise

and most efficient.


ignore slow imprecise

after propagation slow precise

during propagation fast precise




–Directionality







Field reference representation

p.f

r

L1.f

pL1

Field-sensitive

p.f

r

*.f

pL1

Field-based

p.f

r

L1.*

pL1

Field-insensitive

Idea: merge yellow nodes with same abstract object (resp. same field).

L2.f L2.*


L1.f *.fL1.*

r

*.f

L2.f

L1.g

L2.*

*.g

Example (field-based)

L2

p


q = p;L2: r = new O();

p.f = r;t = bar( q );

}


L1

qL1

rL2

*.fL2


return s.f;}

sL1

tL2

Overall (field-based) algorithm

merge each SCC in assignment graph into a single nodetopologically sort resulting DAGfor each node v1 in topological order {for each edge v1 -> v2 {

propagate pt(v1) into pt(v2)}

}

Each edge is processed only once.


Each edge is processed only once.

Worst-case O(n^2).

Also, worst-case is linear in size of output.

In contrast, field-sensitive algorithm is O(n^3).

Example of precision lossA x, y, z;B u, v, w;

A1: x = new A();y = x;

A2: z = new A();B: u = new B();

x.f = u;v = y.f;w = z.f;

B

x

x.f

A1

A1

yA1

uB

y.f

A1.f

z

A2

A2

z.f

A2.fB

B

uB

*.fB


y.f

v

z.f

wB

v wB B

Field-sensitive Field-based

Field sensitivity summary

Java C

field-insensitive sound

slow

imprecise

sound

slow

imprecise

field-based sound

fast

unsound


fast

imprecise

field-sensitive sound

slowest

precise

sound

slowest

precise

Comparison: field-based PTA vs. 0CFA

Field-based subset-based points-to analysis:

pt : Var→℘(Obj)

0CFA:

⇒ Σ = Call × (Var→℘(Lam))





–Directionality







DirectionalityObject x, y, z;

L1: x = new Object();L2: y = new Object();

if(*) {z = x;

} else {z = y;

}

x

L1

L1

zL1 L2

yL2

L2


Subset-based analysis

aka Inclusion-based analysis

aka Andersen's analysis

DirectionalityObject x, y, z;


if(*) {z = x;

} else {z = y;

}

x

L1

L1 L2

zL1 L2

yL1 L2

L2


Equality-based analysis

aka Unification-based analysis

aka Steensgaard's analysis

Implementation of unificationObject x, y, z;


if(*) {z = x;

} else {z = y;

}

x

L1

L1

z

yL2

L2


Step 1: Process allocation edges






if(*) {z = x;

} else {z = y;

}

x

L1

L1

z

yL2

L2



Step 2: Repeatedly unify nodes

connected by assignments

Running time: almost linear






if(*) {z = x;

} else {z = y;

}

x

L1

L1

z

yL2

L2




connected by assignments.






if(*) {z = x;

} else {z = y;

}

x

L1

z

y

L2

L2L1




connected by assignments.

Also unify nodes pointed-to by

same node.

Running time: almost linear




A C examplea = &b;b = &c;a = &d;d = &e;

ab&b

d&d

c&c

e&e




ab&b

d&d

c&c

e&e


&b

&d

&c

&e


ba


&b

d&d

c&c

e&e



ba


&b

d&d

c&c

e&e&b



ba


&b

d&d

c&c

e&e&b

&c



ba


&b

d&d

c&c

e&e


&b

&d

&c


Invariant: each node points to at most one other node.

ba


&b

d&d

c&c

e&e&b

&d

&c

&e



A problem with unification and OOPclass A extends Object {public A() {

super();}

}class B extends Object {...

}L1: x = new A();L2: y = new B();

L2

x

L1

L1

yL2

A().this B().this

Object().this

L1 L2

L2L1


L2: y = new B();



super();}


}L1: x = new A();L2: y = new B();

L2

x

L1

L1

yL2

A().this B().this

Object().this

L1 L2


L2: y = new B();



super();}


}L1: x = new A();L2: y = new B();

L2

x

L1

L1

yL2

A().this B().this

Object().this

L2L1


L2: y = new B();


yx


super();}


}L1: x = new A();L2: y = new B();

L2L1

L1 L2

A().this B().this

Object().this

L2L1L2 L1


L2: y = new B();


yx


super();}


}L1: x = new A();L2: y = new B();

L2L1

L1 L2

A().this B().this

Object().this

L2L1L2 L1


L2: y = new B();

Every pointer points to every object!

... but context sensitivity will fix this.





–Directionality







How big is the call graph?

• Number of methods actually executed: ??

public class Hello {public static final void main(String[] args) {

System.out.println("Hello");}

}

• Number of methods actually executed: ??

• Number of methods in static call graph: ??



• Number of methods actually executed: 498



}


• Number of methods in static call graph: ??






}


• Number of methods in static call graph: 3204


Call graph construction

To determine ... ... we need to know

points-to sets pointer assignment edges

pointer assignment edges reachable methods

call graph edges

reachable methods call graph edges



call graph edges points-to sets

Call graph constructionTo determine ... ... we need to know

points-to sets pointer assignment edges

pointer assignment edges reachable methods

call graph edges


call graph edges points-to setsreachable call graph


points-to

sets

pointer

assignment

edges

reachable

methods

call graph

edges

Ahead of time call graph construction1. Assume every pointer can point to

any object compatible with its

declared type.

2. Explore call graph using this

assumption, listing reachable

methods (Class Hierarchy Analysis).

3. Generate pointer assignment graph

using resulting call edges and

reachable

methods

call graph

edges

fully imprecise

points-to sets

using resulting call edges and

reachable methods.

4. Propagate points-to sets along

pointer assignment graph.


points-to

sets

pointer

assignment

edges

Ahead of time call graph construction1. Assume every pointer can point to

any object compatible with its

declared type.

2. Explore call graph using this

assumption, listing reachable

methods (Class Hierarchy Analysis).

3. Generate pointer assignment graph

using resulting call edges and using resulting call edges and

reachable methods.

4. Propagate points-to sets along

pointer assignment graph.


•no iteration

•very imprecise due to

many reachable

methods

On-the-fly call graph construction1. Start with only initial reachable

methods, no call edges, no pointer

assignment edges, and no points-to

sets.

2. Iteratively generate pointer

assignment edges, points-to sets, call

edges, and reachable methods

implied by current information. reachable call graph implied by current information.

3. Stop when overall fixed point is

reached.


points-to

sets

pointer

assignment

edges

reachable

methods

call graph

edges

On-the-fly call graph construction1. Start with only initial reachable

methods, no call edges, no pointer

assignment edges, and no points-to

sets.

2. Iteratively generate pointer

assignment edges, points-to sets, call

edges, and reachable methods

implied by current information.implied by current information.

3. Stop when overall fixed point is

reached.


•requires iteration

•slower

•more complicated

•much more precise

due to fewer reachable

methods

Partly on-the-fly call graph construction1. Assume all methods are reachable.

2. Generate pointer assignment edges for all methods.

3. Iteratively propagate points-to sets, add call edges, and generate

pointer assignment edges for new call edges.

reachable call graph all methods


points-to

sets

pointer

assignment

edges

reachable

methods

call graph

edges

all methods

reachable

Partly on-the-fly call graph construction1. Assume all methods are reachable.

2. Generate pointer assignment edges for all methods.

3. Iteratively propagate points-to sets, add call edges, and generate

pointer assignment edges for new call edges.

• requires iteration

• speed is in between ahead-of-time and on-the-fly

• complexity is in between...

• still imprecise due to many reachable methods





–Directionality







Context sensitivity motivationObject id(Object o) {

return o;}

void f() {L1: Object a = new Object();L2: Object b = new Object();

Object c = id(a);Object d = id(b);

}

a

L1

L1

id.oL1

b

L2

L2

L2


}

id.retL1 L2

dL1 L2

cL1 L2

Call strings approach (aka cloning)Object id(Object o) {

return o;}

void f() {L1: Object a = new Object();L2: Object b = new Object();L3: Object c = id(a);L4: Object d = id(b);}

a

L1

L1

L3:id.oL1

b

L2

L2

L4:id.oL2


}

L3:id.retL1

dL2

cL1

L4:id.retL2

Summary-based approachObject id(Object o) {

return o;}

void f() {L1: Object a = new Object();L2: Object b = new Object();

Object c = id(a);Object d = id(b);

}

a

L1

L1

b

L2

L2

id()

summary

cL1

dL2


}L1 L2

Challenge: how to design a summary that

•precisely models all effects of id() (and its transitive callees)

•is cheap to compute and represent

•is cheap to instantiate


Field-sensitive subset-based 1-call-site-sensitive

points-to analysis:

pt : (Call × Var ∪ (Obj × Field)) →℘(Obj)

1CFA:

Σ =

Call×(Var→Addr)×(Addr→℘(Lam × Var→Addr))×Call


Limitation of single call site contextObject id(Object o) {L5: return id2(o);}

Object id2(Object o) {return o;

}

void f() {L1: Object a = new Object();

a

L1

L1

L3:id.oL1

b

L2

L2

L4:id.oL2

L5:id2.oL1 L2


L1: Object a = new Object();L2: Object b = new Object();L3: Object c = id(a);L4: Object d = id(b);}

L3:id.ret

dc

L4:id.ret

L1

L5:id2.ret

L2

L1 L2

L1 L2 L1 L2

L1 L2L1 L2

2-call-site context sensitivityObject id(Object o) {L5: return id2(o);}

Object id2(Object o) {return o;

}

void f() {L1: Object a = new Object();

a

L1

L1

L3:id.oL1

b

L2

L2

L4:id.oL2

L3.L5:id2.oL1

L4.L5:id2.oL2


L1: Object a = new Object();L2: Object b = new Object();L3: Object c = id(a);L4: Object d = id(b);}

L3:id.ret

dc

L4:id.ret

L1

L3.L5:id2.retL1

L1 L2

L2L1

L4.L5:id2.ret

L2

L2

k-call-site context sensitivity

• k-Call-Site: keep last k call sites

– space/time complexity exponential in k

• Full Call String: keep full string of call sites

– exponential number of call strings

– recursion: infinite number of call strings

• exclude any call site that is in a recursive cycle from string• exclude any call site that is in a recursive cycle from string

• still exponential

• what if many call sites are in recursive cycle ?

• C: call graph is almost DAG => works well

• Java: half of call graph is one big SCC => no precision


Context-sensitive heap abstractionObject alloc() {L1: return new Object();}

void f() {Object a = alloc();Object b = alloc();

}a

L1

L1

bL1

alloc.retL1



void f() {L2: Object a = alloc();L3: Object b = alloc();}

a

L1

L1

bL1

L2:alloc.retL1

L3:alloc.retL1



void f() {L2: Object a = alloc();L3: Object b = alloc();}

a

L2:L1

b

L2:alloc.retL2:L1

L3:alloc.ret

L3:L1

L3:L1

L3:L1L2:L1



Field-sensitive subset-based 1-call-site-sensitive

points-to analysis with context-sensitive heap

abstraction:

pt :

(Call × Var ∪ (Call × Obj × Field)) →℘(Call × Obj)

1CFA:

Σ =

Call×(Var→Addr)×(Addr→℘(Lam × Var→Addr))×Call


Object-sensitive analysisclass Container {

private Item item;public void set(Item i) {

this.item = i;}

}

L1: Container c1 = new Container();L2: Item i1 = new Item();L3: c1.set(i1);

c1

L1

L1

set.thisL1

i1

L2

L2

set.iL2

set.this.itemL1.itemL2

c2

L4

L4

L4

i2

L5

L5

L5

L4.itemL5 L2 L5


L3: c1.set(i1);


L2 L5 L2 L5

[Milanova&Ryder]

Object-sensitive analysisclass Container {

private Item item;public void set(Item i) {

this.item = i;}

}


c1

L1

L1

L1:set.thisL1

i1

L2

L2

L1:set.iL2

L1:set.this.itemL1.itemL2


L3: c1.set(i1);


L2

c2

L4

L4

L4:set.thisL4

i2

L5

L5

L4:set.iL5

L4:set.this.itemL4.itemL5[Milanova&Ryder]

Object-sensitive analysis

• Like call-site context-sensitive analysis:

– can use strings of abstract objects

– can make heap abstraction (object-)context-sensitive

• Call-site CS and object-sensitive CS have

incomparable precision (neither is theoretically

more precise)more precise)

• In practice, for OO programs, object sensitivity

more precise than call-site sensitivity for the same

context string length


[Milanova&Ryder]

Effect of context-sensitivity in Java

• For call graph construction:

– context sensitivity has some effect

• For cast safety analysis:

– context sensitivity substantially improves precision

– object sensitivity more precise than call sites

– context sensitive heap abstraction further improves – context sensitive heap abstraction further improves

precision

– context strings longer than 1 add little precision

–∞-call-site ignoring cycles less precise than 1-call-site


Context-sensitive equality-based analysisf(a, b, c) {

d = a;d = b;e = c;e = a.f;return e;

}

L1: x = new A();L2: y = new A();

a b c

d e a.f

ret


L2: y = new A();L3: z = new A();L4: w = new A();

x.f = w;v = f(x, y, z);

[Lattner&Adve]



}

L1: x = new A();L2: y = new A();

a b c

d e a.f

ret

L1 L2 L3 L4



x.f = w;v = f(x, y, z);

xL1

yL2

zL3

wL4

x.f

v

[Lattner&Adve]



}

L1: x = new A();L2: y = new A();

a b c

d e a.f

ret

L1 L2 L3 L4



x.f = w;v = f(x, y, z);

xL1

yL2

zL3

wL4

x.fL1.f

v

[Lattner&Adve]



}

L1: x = new A();L2: y = new A();

a b c

d e a.f

ret

L1 L2 L3 L4



x.f = w;v = f(x, y, z);

xL1

yL2

zL3

wL4

x.fL1.fa b c

a.f

ret

v

[Lattner&Adve]



}

L1: x = new A();L2: y = new A();

a b c

d e a.f

ret

L1 L2 L3 L4

a b c

a.f

ret



x.f = w;v = f(x, y, z);

xL1

yL2

zL3

wL4

x.fL1.f

v

[Lattner&Adve]



}

L1: x = new A();L2: y = new A();

a b c

d e a.f

ret

L1 L2 L3 L4

x y

a b c

a.f

ret



x.f = w;v = f(x, y, z);

L1 L2

zL3

wL4

x.fL1.f

v

[Lattner&Adve]



}

L1: x = new A();L2: y = new A();

a b c

d e a.f

ret

L1 L2 L3 L4

x y

a b c

a.f

ret



x.f = w;v = f(x, y, z);

L1 L2

zL3

wL4

x.fL1.f

v

L2.f

[Lattner&Adve]



}

L1: x = new A();L2: y = new A();

a b c

d e a.f

ret

L1 L2 L3 L4

z wx y

a b c

a.f

ret



x.f = w;v = f(x, y, z);

L1 L2 L3 L4

x.fL1.f

v

L2.f

[Lattner&Adve]

Refinement demand-driven analysisObject id(Object o) {

return o;}


a

L1

id.o

b

L2

(3 (4


}

id.ret

dc

)3 )4

[Sridharan&Bodík]


return o;}


a

L1

id.o

b

L2

(3 (4


}

id.ret

dc

)3 )4

L1 ∈ pt(d) ⇔∃ balanced-parens path from L1 to d

[Sridharan&Bodík]


return o;}


a

L1

id.o

b

L2

(3 (4


}

id.ret

dc

)3 )4


But there are lots of paths to search.

[Sridharan&Bodík]


return o;}


a

L1

id.o

b

L2

(3 (4


}

id.ret

dc

)3 )4



Add shortcut edges to graph.

No path even with shortcuts ⇒ no path at all.

[Sridharan&Bodík]


return o;}


a

L1

id.o

b

L2

(3 (4


}

id.ret

dc

)3 )4



Add shortcut edges to graph.

No path even with shortcuts ⇒ no path at all.

On balanced paths found, gradually remove shortcuts.

[Sridharan&Bodík]




–Directionality







Flow sensitivity

L1: a = new Object(); L2: b = new Object();L3: c = new Object();L4: a = b;L5: b = a;L6: c = b;

L1 L2 L3


a b cL3L2L1L2L1L2L1

Flow sensitivity

L1: a = new Object(); a -> {L1}L2: b = new Object(); a -> {L1}, b -> {L2}L3: c = new Object(); a -> {L1}, b -> {L2}, c -> {L3}L4: a = b; a -> {L2}, b -> {L2}, c -> {L3}L5: b = a; a -> {L2}, b -> {L2}, c -> {L3}L6: c = b; a -> {L2}, b -> {L2}, c -> {L2}

Strong updates: overwrite existing pt-set contents

L1 L2 L3


Strong updates: overwrite existing pt-set contentsa b c

L3L2L1L2L1L2L1

Flow sensitivity using SSA forma b c

L1: a = new Object(); a -> {L1} 1L2: b = new Object(); a -> {L1}, b -> {L2} 1 2L3: c = new Object(); a -> {L1}, b -> {L2}, c -> {L3} 1 2 3L4: a = b; a -> {L2}, b -> {L2}, c -> {L3} 4 2 3L5: b = a; a -> {L2}, b -> {L2}, c -> {L3} 4 5 3L6: c = b; a -> {L2}, b -> {L2}, c -> {L2} 4 5 6

L1 L2 L3Currently live reaching


a b cL3L2L1L2L1L2L1

Currently live reaching

definition of each

variable.

Flow sensitivity using SSA forma b c

L1: a1= new Object(); a1-> {L1} 1L2: b2= new Object(); a1-> {L1}, b2-> {L2} 1 2L3: c3= new Object(); a1-> {L1}, b2-> {L2}, c3-> {L3} 1 2 3L4: a4= b2; a4-> {L2}, b2-> {L2}, c3-> {L3} 4 2 3L5: b5= a4; a4-> {L2}, b5-> {L2}, c3-> {L3} 4 5 3L6: c6= b5; a4-> {L2}, b5-> {L2}, c6-> {L2} 4 5 6

L1 L2 L3L1 L2 L3


a1 b2 c3L3L2L1

a4 b5 c6L2L2 L2

For local variables, FI analysis on SSA

form gives same result as FS analysis

on original program.

a b cL3L2L1L2L1L2L1

Flow sensitivity using SSA forma

L1: if(*) {L2: a = b; 2L3: } else {L4: a = c; 4L5: } ?L6: ?L7: d = a; ?


Which definition of a is current at L7?

Flow sensitivity using SSA forma

L1: if(*) {L2: a = b; 2L3: } else {L4: a = c; 4L5: } ?L6: a = φ(a2,a4) 6L7: d = a; 6


Which definition of a is current at L7?

Use φ to create new definition of a.

pt(a6) = pt(a2) ∪ pt(a4)

When does flow sensitivity matter?

Java C/C++

local variables no, use SSA form

address-taken local variables no, don't exist possibly

global variables unlikely, values usually long-lived

fields of heap objects no strong updates on heap objects

unless analysis extended with


unless analysis extended with

single-concrete-object abstraction




–Directionality







a tour of pointer analysis - korea...

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